# Chinese remainder theorem

Let $R$ be a commutative ring with identity. If $I_{1},\ldots,I_{n}$ are ideals of $R$ such that $I_{i}+I_{j}=R$ whenever $i\neq j$, then let

 $I=\cap_{i=1}^{n}I_{i}=\prod_{i=1}^{n}I_{i}.$

The sum of quotient maps $R/I\to R/I_{i}$ gives an isomorphism

 $R/I\cong\prod_{i=1}^{n}{R}/{I_{i}}.$

This has the slightly weaker consequence that given a system of congruences $x\cong a_{i}\pmod{I_{i}}$, there is a solution in $R$ which is unique mod $I$, as the theorem is usually stated for the integers.

Title Chinese remainder theorem ChineseRemainderTheorem1 2013-03-22 12:16:43 2013-03-22 12:16:43 bwebste (988) bwebste (988) 7 bwebste (988) Theorem msc 11N99 msc 11A05 msc 13A15 ChineseRemainderTheoremInTermsOfDivisorTheory